Opening the AirPods Pro case is going to be nearly impossible; according to this takedown: https://www.ifixit.com/News/66194/apples-new-airpods-are-telling-users-to-replace-the-batteries-already-too-bad-thats-impossible
It sounds like the battery came physically disconnected from the case's main board.
So, replacing the battery, or even opening up the case, is not recommended. You can try to reset the AirPods first. (With the case plugged in, press and hold the back button for fifteen seconds, or until the LED flashes amber)
But, I would say it is worth it to go ahead and get a new case from Apple, because the AirPods are useless without the case. Plus, Apple just released the third generation of AirPods Pro (AirPods Pro 2nd Generation, USB-C). And this new case will work fine with your AirPods. [link|https://www.apple.com/shop/product/MUYG3AM/A/magsafe-charging-case-usb%E2%80%91c-for-airpods-pro-2nd-generation?fnode=b6f9f83c6968dd35df4ce9f281920f49cf2343afe2022ce4ea339305965a5b4899d9d2461a7bac4ce45fdf7bbe2142c7c25d11f3707a865a23d11516bb4cb47ee140a0b9c17b475accdf9759379568ede4f01498ee5b0944ba5217abc74c9aa91ba2916c683e0581366dd095ff155a2e815a47e016016ae92458871d61050afb&fs=f%3Dheadphoneacc%26fh%3D47d1%252Be62115|AirPods Pro Case (USB-C)|new_window=true]